Social Science Reasoning
Using Statistics
Psychology 138
2015
• Exam 2 is Wed. March 4th
• Quiz 4
– Fri Feb 27th
– Covers z-scores, Normal distribution, and describing
correlations
Announcements
Reasoning in Psychology
Using Statistics
• Transformations: z-scores
• Normal Distribution
• Using Unit Normal Table
– Today’s lecture puts lots of
stuff together:
•
•
•
•
Probability
Frequency distribution tables
Histograms
Z-scores
Outline
Reasoning in Psychology
Using Statistics
Today
Start the quincnux machine
HHH
Number of heads
3
HHT
2
HTH
2
HTT
1
THH
2
THT
1
TTH
1
TTT
0
Flipping a coin example
Reasoning in Psychology
Using Statistics
probability
Number of heads
3
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
2
X
f
p
3
1
.125
2
2
1
3
3
.375
.375
1
0
1
.125
1
2
1
0
Flipping a coin example
Reasoning in Psychology
Using Statistics
probability
What’s the probability of
flipping three heads in a
row?
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
p = 0.125
Think about the area under the
curve as reflecting the
proportion/probability of a
particular outcome
Flipping a coin example
Reasoning in Psychology
Using Statistics
probability
What’s the probability of
flipping at least two heads
in three tosses?
.4
.3
.2
.1 .125
.375 .375 .125
p = 0.375 + 0.125 = 0.50
0 1 2 3
Number of heads
Flipping a coin example
Reasoning in Psychology
Using Statistics
probability
What’s the probability of
flipping all heads or all tails
in three tosses?
.4
.3
.2
.1 .125
.375 .375 .125
p = 0.125 + 0.125 = 0.25
0 1 2 3
Number of heads
Flipping a coin example
Reasoning in Psychology
Using Statistics
• Theoretical distribution (The “Bell Curve”)
– Commonly occurs in nature
– Common approximate empirical distribution for deviations around a continually scaled
variable (errors of measurement)
• Defined by density function (area under
curve) for variable X given μ & σ2
• Symmetrical & unimodal; Mean = median =
mode
• ±1 σ are inflection points of curve (change of
direction)
f(x; μ,
Normal Distribution
Reasoning in Psychology
Using Statistics
σ2 )
=
1
2ps 2
e -(X -m )
2
f (z)= 1 e-z/2
2p
Check out the
quincnux machine
/ 2s 2
• Theoretical distribution (The “Bell Curve”)
• Use calculus to find areas under curve (rather than frequency of a
score)
• We will use a table
rather to find the
probabilities rather
than do the calculus.
f(x; μ,
Normal Distribution
Reasoning in Psychology
Using Statistics
σ2 )
=
1
2ps 2
e -(X -m )
f (z)= 1 e-z/2
2p
2
/ 2s 2
• Theoretical distribution (The “Bell Curve”)
– Important landmarks in the distribution (and the areas under the curve)
• %(μ to 1σ) = 34.13
• p(μ < X < 1σ) + p(μ > X > -1σ) ≈ .68
• %(1σ to 2σ) = 13.59
• p(1σ < X < 2σ) + p(-1σ > X > -2σ) = 27%, cumulative = .95
• %(2σ to ∞) = 2.28
• p(X > 2σ) + p(X < -2σ) ≈ 5%, cumulative = 1.00
68%
34.13
95%
100%
2.28
13.59
Normal Distribution
Reasoning in Psychology
Using Statistics
• We will use the
unit normal table
rather to find other
probabilities
• Lots of places to get the Unit Normal Table information
» But be aware that there are many ways to organize the table, it is
important to understand the table that you use
– Unit normal table in your reading packet
– And online:
http://psychology.illinoisstate.edu/jccutti/psych138/resources copy/TABLES.HTMl - ztable
– “Area Under Normal Curve” Excel tool
•(created by Dr. Joel Schneider)
– Bell Curve iPhone app
– Do a search on “Normal Table” in Google.
Resources and tools
Reasoning in Psychology
Using Statistics
z
p
|z|
.00
.01
0
:
:
1.0
:
2.0
:
3.0
0.5000
:
:
0.1587
:
0.0228
:
0.0013
0.4960
:
:
0.1562
:
0.0222
:
0.0013
• Proportions beyond z-scores
• Same p-values for + and - z-scores
• p-values = 0.50 to 0.0013
For z = 1.00 , 15.87% beyond, p(z > 1.00) = 0.1587
1.01, 15.62% beyond, p(z > 1.00) = 0.1562
Note. : indicates skipped rows
Unit Normal Table
(in reading packet and online)
Reasoning in Psychology
Using Statistics
|z|
.00
.01
0
:
:
1.0
:
2.0
:
3.0
0.5000
:
:
0.1587
:
0.0228
:
0.0013
0.4960
:
:
0.1562
:
0.0222
:
0.0013
• Proportions beyond z-scores
• Same p-values for + and - z-scores
• p-values = 0.50 to 0.0013
For z = -1.00, 15.87% beyond, p(z < -1.00) = 0.1587
Note. : indicates skipped rows
Unit Normal Table
Reasoning in Psychology
Using Statistics
(in reading packet)
|z|
.00
.01
0
:
:
1.0
:
2.0
:
3.0
0.5000
:
:
0.1587
:
0.0228
:
0.0013
0.4960
:
:
0.1562
:
0.0222
:
0.0013
• Proportions beyond z-scores
• Same p-values for + and - z-scores
• p-values = 0.50 to 0.0013
For z = 2.00, 2.28% beyond, p(z > 2.00) = 0.0228
2.01, 2.22% beyond, p(z > 2.01) = 0.0222
As z increases, p decreases
Note. : indicates skipped rows
Unit Normal Table
(in reading packet)
Reasoning in Psychology
Using Statistics
|z|
.00
.01
0
:
:
1.0
:
2.0
:
3.0
0.5000
:
:
0.1587
:
0.0228
:
0.0013
0.4960
:
:
0.1562
:
0.0222
:
0.0013
• Proportions beyond z-scores
• Same p-values for + and - z-scores
• p-values = 0.50 to 0.0013
For z = -2.00, 2.28% beyond, p(z < -2.00) = 0.0228
Note. : indicates skipped rows
Unit Normal Table
(in reading packet)
Reasoning in Psychology
Using Statistics
z
.00
.01
-3.4
:
:
-1.0
:
0
:
1.0
:
:
3.4
0.0003
:
:
0.1587
:
0.5000
:
0.8413
:
:
0.9997
0.0003
:
:
0.1562
:
0.5040
:
0.8438
:
:
0.9997
• Proportions left of z-scores: cumulative
• Requires table twice as long
• p-values 0.0003 to 0.9997
For z = +1, 50% + 34.13
= 84.13% to the left
Cumulative % of population
starting with the lowest value
Unit Normal Table
cumulative version (in some other books)
Reasoning in Psychology
Using Statistics
• Population parameters of SAT:
μ = 500, σ = 100, normally distributed
Example 1
Suppose you got 630 on SAT. What % who take SAT get
your score or better?
z=
X -m
s
From table:
630 - 500
=
=1.3
100
z(1.3) =.0968
9.68% above
this score
1. Solve for z-value of 630.
2. Find proportion of normal distribution above that value.
Hint: I strongly suggest that you sketch the problem to check your answer against
SAT examples
Reasoning in Psychology
Using Statistics
• Population parameters of SAT:
μ = 500,  = 100, normally distributed
Example 2
Suppose you got 630 on SAT. What % who take SAT get
your score or worse?
z=
X -m
s
=
630 - 500
=1.3
100
From table:
z(1.3) =.0968
100% - 9.68% = 90.32% below
score (percentile)
1. Solve for z-value of 630.
2. Find proportion of normal distribution below that value.
SAT examples
Reasoning in Psychology
Using Statistics
• In lab
– Using the normal distribution
• Questions?
Check the quincnux machine
Wrap up
Reasoning in Psychology
Using Statistics